A mathematical approach to Wick rotations
by
Christer Helleland
Thesis submitted in fulfilment of the requirements for the degree of
PHILOSOPHIAE DOCTOR (PhD)
!
Faculty of Science and Technology Department of Mathematics and Physics
2019
NO-4036 Stavanger NORWAY
www.uis.no
©2019 Christer Helleland ISBN: Click to enter ISBN.
ISSN: Click to enter ISSN.
PhD: Thesis UiS No. Click to enter PhD No.
List of Tables 6
Acknowledgements 8
Part 1. Introduction 9
Part 2. Preliminaries 12
1. Cartan involutions of linear Lie groups 12
2. Geometric invariant theory (GIT) 16
Part 3. Wick-rotations and real GIT 22
1. Introduction 22
2. Mathematical Preliminaries 23
2.1. Real form of a complex vector space 23
2.2. Real slices 24
2.3. Compatible real forms 25
3. Holomorphic Riemannian manifolds 27
3.1. Complexification of real manifolds 27
3.2. Complex differential geometry 28
3.3. Real slices from a frame-bundle perspective 29
4. Lie groups 31
4.1. Complex Lie groups and their real forms 31
4.2. Example: Split G2-holonomy manifolds 32
5. A standard Wick-rotation to a real Riemannian space 34 5.1. Minimal vectors and closure of real semi-simple orbits 34 5.2. Compatible triples and intersection of real orbits 36
5.3. The real Riemannian case 37
5.4. The adjoint action of the Lorentz groups O(n−1,1) 38 5.5. Uniqueness of real orbits and the class of complex Lie groups 41 6. Applications to the pseudo-Riemannian setting 45
6.1. Pseudo-Riemannian examples 47
Acknowledgements 48
Appendix A. On compatible Hermitian inner products 48 Appendix B. The boost-weight decomposition in the Lorentzian case 49 Part 4. A Wick-rotatable metric is purely electric 51
3
1. Introduction 51
2. The electric/magnetic parts of a tensor 53
3. The Riemann curvature operator 54
4. Discussion 58
Acknowledgements 59
Part 5. Real GIT with applications to compatible representations
and Wick-rotations 60
1. Introduction 60
2. Mathematical Preliminaries 61
2.1. Real slices and compatibility 61
2.2. A Wick-rotation implies a standard Wick-rotation 62
2.3. Real GIT for semi-simple groups 65
2.4. Real GIT for linearly real reductive groups 67
3. Balanced representations 71
4. Compatible representations 75
5. Compatible real orbits 77
6. Applications to Wick-rotations of arbitrary signatures 80
6.1. The isometry action ofO(n,C) on tensors 80
6.2. Purely electric/magnetic spaces 82
6.3. Invariance theorem for Wick-rotation at a point p 83 6.4. A note on Wick-rotations of the same signatures 84
6.5. Wick-rotatable metrics 85
6.6. Universal metrics 86
6.7. On the set of tensors with identical invariants 87
Acknowledgements 90
Part 6. Wick-rotations of pseudo-Riemannian Lie groups 91
1. Introduction 91
2. Preliminaries 92
2.1. Real forms and left-invariant metrics 92
2.2. Wick-rotations of pseudo-Riemannian manifolds 94
2.3. Real GIT on compatible representations 97
2.4. The isometry action on bilinear forms into the Lie algebra 100 2.5. Wick-rotatable tensors of pseudo-Riemannian manifolds 101
3. An invariant of Wick-rotation of Lie groups 104
4. Conjugacy of Cartan involutions 110
5. Wick-rotating a Lorentzian signature 112
6. A remark on Wick-rotatable tensors of Lie groups 114
7. Wick-rotating an algebraic soliton 116
Part 7. Holomorphic inner product spaces on Lie algebras 119
1. Real slices up to isomorphism 119
2. On the existence of a compact real form 120
Part 8. The bibliography 126
References 126
List of Tables
1 The relation between Riemann types and the vanishing of boost weight components. For example, (R)+2 corresponds to the frame components
R1i1j. 50
Abstract. In this thesis we define Wick-rotations mathematically using pseudo- Riemannian geometry, and relate Wick-rotations to real geometric invariant theory (GIT). We discover some new results concerning the existence of Wick- rotations (of various signatures). For instance we show that a Wick-rotation of a pseudo-Riemannian space (at a fixed pointp) to a Riemannian space forces the space to beRiemann purely electric (RPE). We also definecompatibility among representations and relate them to real GIT and Wick-rotations. The polyno- mial curvature invariants of pseudo-Riemannian spaces are also considered and related to Wick-rotations.
Wick-rotations of a special class of pseudo-Riemannian manifolds (M, g) are also studied; namely Lie groups G equipped with left-invariant metrics. We prove some new results concerning the existence of real slices (of Lie algebras) of certain signatures of a holomorphic inner product space (gC, gC) (on a complex Lie algebra). The definition of a Cartan involution for a semisimple Lie algebra is defined for a general Lie algebra equipped with a pseudo-inner product: (g, g), and the theorems of Cartan (concerning Cartan involutions) are generalised and proved. For instance we prove that a pseudo-Riemannian Lie group (G, g) can be Wick-rotated to a Riemannian Lie group ( ˜G,˜g) if and only if there exist a Cartan involution of the Lie algebrag.
Acknowledgements
I would like to thank the University of Stavanger (UiS), and my main advisor Sigbjørn Hervik for guiding me through the Ph.D project; thanks for always being available for whatever questions I had, and for letting me pursue the directions I wanted. Also a thanks to my second advisor Boris Kruglikov for questions I had during a stay at the University of Tromsø.
Part 1. Introduction
Wick-rotations arise from physics, named afterGian Carlo Wick, and is a math- ematical trick of reducing a problem in Minkowski space to a problem of Euclidean space. This is done by transforming the Minkowski metric (which is a Lorentzian metric) to a Euclidean metric, and often the problem reduced to the Euclidean case will be easier to solve. Thus Wick-rotations are for instance of interest in fields like quantum mechanics, statistical mechanics and Euclidean gravity and thereon. One can ask of the limitations of such Wick-rotations, for when one can preform such a trick, and for a general spacetime one can ask the question if such a Wick-rotation to a Euclidean space is even possible.
In this thesis we consider a mathematical approach to Wick-rotations in the framework of pseudo-Riemannian geometry. A motivation for our approach also comes from the theory of Lie groups, i.e the example of a complex semisimple Lie group GC equipped with its Killing form −κ(·,·). The space (GC,−κ) is a holomorphic Riemannian space, and the real forms G ⊂ GC give rise to pseudo- Riemannian spaces with real-valued metrics at the identity restricted from the complex Killing form. The real metrics are then the real Killing forms on the Lie algebrag⊂gC. Moreover a compact real form U ⊂GC gives rise to a Riemannian space in this way.
(GC,−κ) ←−−−−−−−Riemannian (U,−κ|u) x
P seudo (G,−κ|g)
Thus we consider a general pseudo-Riemannian space (M, g) of some signature (p, q), and consider the question: When does there exist a Riemannian manifold ( ˜M,g)˜ Wick-rotated to(M, g)?
More generally we consider an arbitrary pseudo-Riemannian space ( ˜M,g) (not˜ necessarily Riemannian) of some (possibly different) signature (˜p,q) and ask the˜ question of when it can be Wick-rotated to (M, g). An interesting special case to consider are pseudo-Riemannian Lie groups (G, g), i.e Lie groups equipped with left-invariant metrics (not necessarily semisimple groups).
We are thus interested in finding necessary or sufficient conditions that enables a Wick-rotation to different signatures, with a special interest in the case of Wick- rotating to a Riemannian space.
A mathematical motivation for studying Wick-rotations in this thesis also comes from the classification of pseudo-Riemannian spaces, and that of the polynomial curvature invariants of pseudo-Riemannian spaces. At a point p ∈ M, these invariants are special polynomial invariants of an action of a pseudo-orthogonal group O(p, q) on a tensor space V, restricted to the Riemann tensor R and its covariant derivatives ∇kR up to some kth order viewed as vectors inside V. The
polynomial curvature invariants are related to the Cartan equivalence principle which relates the curvature tensors to the metric.
For example suppose we are given two pseudo-Riemannian spaces (M, g) and ( ˜M,g), and we impose that they have the same polynomial curvature invariants,˜ then we may ask: How are (M, g)and( ˜M,g)˜ related? andUnder what conditions are the metrics Wick-rotated?
The parts of the thesis is structured as follows.
(2) We provide some known concepts and mathematical tools used in the later parts. This part contains no original results.
(3) We define Wick-rotations by considering pseudo-Riemannian manifolds as real slices of a holomorphic Riemannian manifold. From a frame bundle viewpoint Wick-rotations between different pseudo-Riemannian spaces can then be studied through their structure groups which are real forms of the corresponding complexified Lie group (different real forms O(p, q) of the complex Lie group O(n,C)). In this way, we can use real GIT (geometric invariant theory) to derive several new results regarding the existence, and non-existence, of such Wick-rotations. As an explicit example, we Wick rotate a known G2-holonomy manifold to a pseudo-Riemannian manifold with split-G2 holonomy.
(4) We show that a metric of arbitrary dimension and signature which allows for a standard Wick-rotation to a Riemannian metric necessarily has a purely electric Riemann and Weyl tensor.
(5) Motivated by Wick-rotations of pseudo-Riemannian manifolds, we study real geometric invariant theory (GIT) and compatible representations. We extend some of the results from earlier works [20,21], in particular, we give some sufficient as well as necessary conditions for when pseudo-Riemannian manifolds are Wick-rotatable to other signatures. For arbitrary signatures, we consider a Wick-rotatable pseudo-Riemannian manifold with closed O(p, q)-orbits, and thus generalise the existence condition found in [21].
Using these existence conditions we also derive an invariance theorem for Wick-rotations of arbitrary signatures.
(6) We study Wick-rotations of left-invariant metrics on Lie groups, using re- sults from real GIT ([20], [19]). An invariant for Wick-rotation of Lie groups is given, and we describe when a pseudo-Riemannian Lie group (a Lie group with a left-invariant metric) can be Wick-rotated to a Riemann- ian Lie group. We define a Cartan involution of a general Lie algebra, and prove a general version of ´E. Cartan’s result, namely the existence and conjugacy of Cartan involutions.
(7) Continuing with the ideas of Part 6, we consider a holomorphic inner prod- uct space on a complex Lie algebra: (gC, gC). We prove some new results re- garding the existence of a compact real form (of Lie algebras)u⊂(gC, gC), i.eu is a real form with Euclidean signature: gC(u,u)>0.
(8) The bibliography of the thesis.
Part 2. Preliminaries
1. Cartan involutions of linear Lie groups
In this section we introduce some known concepts and tools that we use through- out this thesis. The Lie algebras we speak of in this section, shall be defined over RorC. We shall distinguish by saying real (complex), the same goes for Lie groups.
We recall that any Lie algebra g has a Killing form: κ(·,·), defined by:
κ(x, y) :=tr
ad(x)◦ad(y)
, x, y∈g,
where g −−−→ad(x) g ∈ End(g), is defined by ad(x)(y) := [x, y], x, y ∈ g. The Killing form κ is easily seen to be associative or invariant, i.e it satisfies κ([x, y], z) = κ(x,[y, z]) for all x, y, z ∈g.
Definition 1.1. A Lie algebra is said to besemisimple ifκ(·,·) is non-degenerate.
A Lie group G with Lie algebra g is said to be semisimple if its Lie algebra g is semisimple.
Definition 1.2. A Lie algebra is said to be simple if it is non-abelian and does not have any non-trivial ideals.
A Lie algebra can be shown to be semisimple if and only if it does not have any non-trivial abelian ideals, thus solvable Lie algebras do not belong to this class. It is also a standard result that any semisimple Lie algebra can written into a direct sum of simple ideals [16].
Definition 1.3. A Cartan involution of a semisimple Lie algebra is an involution of Lie algebras: g−→θ g, such that −κ(·, θ(·)) is positive definite.
The eigenspace decomposition w.r.t a Cartan involution is often called aCartan decomposition. As an example consider the semisimple real Lie algebra sl2(R), then the mapx 7→ −xt, is a Cartan involution, and thus a Cartan decomposition is given by:
sl2(R) =so(2)⊕ {x∈sl2(R)|xt=x}.
By a result of Cartan, every semisimple Lie algebra has a Cartan involution, and moreover any two Cartan involutions are conjugate by an inner automorphism (see for example [16]):
Theorem 1.4. Any real semisimple Lie algebra has a Cartan involution which is unique up to conjugation.
By Ado’s theorem every Lie algebra is linear, thus for a general Lie algebra one can define the notion of a Cartan involution:
Definition 1.5. Letg⊂gl(V) be a real Lie subalgebra. Then aCartan involution ofgis an involution of Lie algebras that restricts from an involutiongl(V)−→θ gl(V) such that θ|sl(V) is a Cartan involution of sl(V).
It is a fact that any Cartan involution of sl(V) has the form x 7→ −x∗, where h−,−iis some inner product onV withhx∗(v1), v2i=hv1, x(v2)i for allv1, v2 ∈V. Thus the Cartan involutions of gl(V) also have this form.
One shall note that the definition of a Cartan involution of a semisimple real Lie algebra coincides with the previous definition. Indeed if g ⊂ gl(V) is semisimple and θ is a Cartan involution in the sense of Definition 1.3, then g⊂sl(V) and by using Mostow’s Theorem ([33], Thm 6) one can find a Cartan involution of sl(V) extending θ. Conversely if θ is a Cartan involution of gl(V) leaving g invariant, then it follows thatθ is a Cartan involution ofgin the sense of Definition1.3([16], Ch. IX, Lem. 2.2).
Definition 1.6. A Lie algebra g is said to be reductive if g = [g,g]⊕z(g) where [g,g] is either trivial or a semisimple Lie algebra.
For examplegln(R) is reductive withgln(R) = sln(R)⊕ hIni, and an example of a Cartan involution is the map x7→ −xt.
More generally a Lie subalgebra g ⊂h is said to be reductive in h if the repre- sentation of g:
x·y := [x, y]∈h, x∈g, y ∈h, is completely reducible.
Clearly the real Lie algebras for which a Cartan involution exist belong to the class of reductive Lie algebras. Moreover if θ is a Cartan involution of g, and z(g) =V+⊕V− is the eigenspace decomposition w.r.tθ, then:
−κ(·, θ(·)) +B(·, θ(·))>0
is positive definite, where κ is the Killing form of [g,g] and B is a pseudo-inner product (i.e a non-degenerate symmetric bilinear form) on z(g) of signature (p, q) for p:=Dim(V+) and q :=Dim(V−).
Recall that aquadratic Lie algebra gis a Lie algebra equipped with an invariant pseudo-inner product g(·,·), i.e g is a symmetric non-degenerate bilinear form on g satisfying: g([x, y], z) = g(x,[y, z]),∀x, y, z ∈ g. A Lie algebra g is said to be compact if there exist such ag which is also an inner product, i.eg is also positive definite. For example the class of Lie algebras: g, with a Cartan involutionθ = 1, are compact; by noting the above pseudo-inner product: −κ(−,−) +B(−,−).
For linear Lie groups one defines:
Definition 1.7. Let G ⊂ GL(V) be a real linear Lie group, then an involution G−→Θ Gof Lie groups is said to be aCartan involution if Θ restricts from a Cartan involution of GL(V) (i.e the differential is a Cartan involution of gl(V)).
If θ is a Cartan involution of a semisimple Lie algebra of a Lie group G (not necessarily linear) then there exists a unique involution G −→Θ G with differential θ.
Thus for a general semisimple Lie group one have the notion of a Cartan invo- lution (see for example [12]):
Theorem 1.8. LetGbe a semisimple real Lie group, and θ a Cartan involution of g=k⊕p. Then there exist a unique involution of Lie groups: G−→Θ G satisfying:
(1) The differential ofΘ at 1∈G is θ, i.e dΘ =θ.
(2) The fix points of Θ denoted by K ⊂ G has Lie algebra k and k ∈ K are those elements of G satisfying [Ad(k), θ] = 0.
(3) K is compact if and only if Z(G0) is finite and G has finitely many con- nected components (fcc).
(4) There is a diffeomorphism K×p→G given by (k, x)7→kex.
For example if we consider the semisimple linear group: SL2(R), then the map g 7→(g−1)t, is a Cartan involution, with fix-pointsK =SO(2). Another example is SL2(C) viewed as a real Lie group, then the mapg 7→(g−1)†is a Cartan involution with fix-pointsK =SU(2).
There are many distinct definitions of a reductive Lie group, however some au- thors define areal reductive Lie groupto be a linear Lie groupG⊂GL(V) together with a Cartan involution Θ (see [29]). For example areal reductive algebraic group G ⊂ GL(V), is a real algebraic group such that g is reductive in gl(V), and it belongs to this class of real reductive Lie groups (see [35]).
Let G ⊂ GL(V) be such a real linear Lie group with a Cartan involution Θ, and θ := dΘ be the Cartan involution of its Lie algebra g ⊂ gl(V). If g = k⊕p denotes the Cartan decomposition w.r.tθ (i.e the eigenspace decomposition), and K ⊂ G is the fix points of Θ, then we can globally write G = Kep, where K is maximally compact and has Lie algebra k. Now any Cartan involution of GL(V) has the form A7→(A−1)∗ where
hA∗(v1), v2i=hv1, A(v2)i,
for some inner product onV. Thus K ⊂O(V,h·,·i), and ep consists of symmetric operators w.r.th·,·i. Therefore such groups Gare also self-adjoint, i.e G∗ =G.
Definition 1.9. LetgC denote a complex Lie algebra, then a real Lie subalgebra g⊂gC is said to be a real form if there exist a conjugation mapσ (i.e σ is a real involution satisfyingσ(ix) =−ix) with fix points g.
Thus as a real Lie algebra we may write: gC = g⊕ig. All the real forms up to isomorphism of a complex semisimple Lie algebra are in bijection with the conjugacy classes of involutions of the complex Lie algebra (see [12]):
{[g]|ga real f orm} → {[θ]|θ an involution}.
The map is well-defined by sending a real form gto a complexified Cartan involu- tion θC. Thus from the complex Killing form perspective: κ(·,·), the existence of an involution θ of gC gives rise to a real form g which is real-valued on κ(·,·), i.e κ(g,g)∈R. The restriction is in fact just the real Killing form ofg.
For example sl2(R) and su(2) are all the real forms (up to isomorphism) of sl2(C); by noting the conjugation maps: x7→x, and¯ x7→ −x†.
Any complex semisimple Lie algebra has a special real form which is unique up to isomorphism, namely a real form u for which the Cartan involution is just θ = 1, also called acompact real form. Thus the restriction of the complex Killing form: −κ(·,·) to u is of Euclidean signature.
For the following result, see for example [16]:
Theorem 1.10. Any complex semisimple Lie algebra has a compact real form.
Moreover up to isomorphism a compact real form is unique.
If u ⊂ gC is a compact real form then the conjugation map τ of u is a Cartan involution of gC viewed as a real Lie algebra. Thus a Cartan decomposition is given by: gC = u⊕iu. Moreover the real Killing form of gC is just 2Re(κ(·,·)), where κ is the complex Killing form and Reis the real part.
One shall also note that if g⊂gC is a semisimple real form, and g=k⊕p is a Cartan decomposition, then k⊕ip is a compact real form of gC.
A simple real Lie algebra g fall into one of the following two classes [40]; either its complexification gC is simple in which case all of its real forms are simple, or g has a complex structureg −→J g, in which case gC ∼= g⊕g where g is viewed as the complex Lie algebra constructed by J. Recall that a complex structure is an endomorphism J ong satisfying J2 =−1 and [J(x), y] =J([x, y]) for all x, y ∈g.
For example viewing the complex simple Lie algebra: sl2(C), as a real Lie algebra denoted: sl2(C)R, then the latter real simple Lie algebra has a complex structure and is a real form of sl2(C)⊕sl2(C).
Definition 1.11. Let G be a real Lie group, then the universal complexification group of G is a pair (GC, η), where GC is a complex Lie group, and G −→η GC is a real Lie homomorphism, satisfying the universal property. This means that for any real Lie homomorphism G −→ψ HC into a complex Lie group HC there is a unique Lie homomorphismGC−→l HC, such that the following diagram commutes:
GC −−−→l HC
η
x
1 x
G −−−→ψ HC (1)
For example consider the complex semisimple linear Lie group SL2(C), then it is the universal complexification group of SU(2) and of the universal covering
group SL^2(R) of SL2(R). Note that SL^2(R) is not linear. Another example is if we view SL2(C) as a real Lie group of dimension 6, and use that SL2(C) is simply connected, then the universal complexification group is just the product:
SL2(C)×SL2(C).
Definition 1.12. ([31]) A linearly complex reductive Lie group GC is a complex Lie group that is the universal complexification of a compact real Lie group G.
If (GC, η) is the universal complexification of a compact group U (i.e GC is lin- early complex reductive) thenη is injective with closed image, thus by identifying U ∼= η(U) ⊂ GC, then U is a compact real form (see defn below), and moreover there is a diffeomorphism [31]:
U ×u→GC, (u, x)7→ueix.
One shall note that the linearly complex reductive Lie groups are precisely the complex reductive algebraic groups, for more details on these groups see for exam- ple [34, 31]. For example a 1-dimensional complex tori C× is a linearly complex reductive Lie group which is the universal complexification of the circle S1.
There are many distinct definitions in the literature of areal form of a complex Lie group, however we shall occasionally use this one [12]:
Definition 1.13. A real Lie subgroupG of a complex Lie group GC is said to be a real form if g is a real form of the Lie algebra of GC, and moreover as a group product we have GC=GGC0 whereGC0 is the identity component.
Note when GC is connected then for a real Lie group G⊂GC to be a real form is just the condition that the Lie algebra g is a real form ofgC.
Definition 1.14. A real formU ⊂GC shall be called a compact real form if U is compact.
For example, if a complex Lie group (with finitely many components (fcc)) have a compact real form then it must belong to the class of linearly complex reductive Lie groups [31]. All the semisimple complex Lie groups belong to this class.
2. Geometric invariant theory (GIT)
Our fields K of interest in this section are C ⊃R. We begin by defining some preliminary concepts (see for example [48]). An affine variety or an algebraic set X shall mean a subsetX ⊂Kn of the form:
X ={x∈Kn|(∀f ∈I)(f(x) = 0)}:=ZK(I),
for some ideal I ⊂ K[X1, . . . , Xn]. Set IK(X) for the set of all the polynomials f ∈K[X1, . . . , Xn] such thatf(X) = 0.
Definition 2.1. An algebraic set X ⊂Cn is said to bedefined over R if IC(X) is generated by real polynomials inR[X1, . . . , Xn].
Thus in such a case the real points X(R) := X∩Rn is a real algebraic subset of Rn, such that the Zariski-closure of X(R) in Cn is X. On the other hand if X ⊂Rn is algebraic then the Zariski-closure ofX inCn, denotedXC (often called a complexification of X) is defined over R, and if IR(X) = hf1, . . . , fki then also IC(XC) = hf1, . . . , fki.
Definition 2.2. Let X, Y be complex algebraic sets both defined over R, and X −→F Y be a morphism. Then F is said to be defined over R if F restricts to a real morphism: X(R)
F|X(R)
−−−→Y(R).
One shall also note that the real polynomial algebra R[X(R)] is a real form of C[X], by noting the conjugation map:
C[X]3f 7→f¯∈C[X].
For a real algebraic group G⊂GL(E) the Zariski-closure in GL(EC) is a com- plex algebraic group GC, and is often called an algebraic complexification of G.
One shall note that G⊂GC is a real form.
If GC is a complex algebraic group defined over R and GC acts on a complex affine variety X, also defined over R, then the action is said to be defined over R if the map GC×X → X is defined over R. Thus in such a case there is a real action of G(R) on X(R), and if C[X]GC denotes the GC-invariant polynomials of the action, then the conjugation map above leaves the algebra invariant, and it is straightforward to show that the corresponding real form is precisely the real polynomial invariants: R[X(R)]G(R).
Geometric invariant theory (GIT) in algebraic geometry (first developed by D.
Mumford) is concerned with the problem when an algebraic group G acts on a variety X, and the question of when there exist a quotient
X
G, X → XG
where
X
G is a variety and X → XG a morphism satisfying certain geometrical properties.
One special case to consider is when a complex algebraic groupGC acts rationally on a complex vector space VC∼=Ck. We shall recall some results over C.
Let GC be a linearly complex reductive Lie group (for example a semi-simple complex Lie group), andGC−→ρC GL(VC) be any representation of Lie groups. Then ρC is also a rational representation w.r.t its unique algebraic group structure. Let C[VC] denote the coordinate ring of polynomials ofVC, and C[VC]GC, denote the subalgebra of GC-invariant polynomials, i.e f ∈C[VC] satisfying
f(g·v) =f(v), ∀g ∈GC, v ∈VC. Then the following theorem hold [49]:
Theorem 2.3 (Hilbert). The polynomial ring of invariants: C[VC]GC, is a finitely generated C-algebra.
LetI :={f1, . . . , fN}be a finite generating set for C[VC]GC, and define a map:
VC−→p CN, v 7→(f1(v), . . . , fN(v)).
Let Y denote its closed image in CN, then Y is an affine variety with coordinate ring: C[Y]∼=C[VC]GC. The map p is a good categorial quotient, i.e the following hold:
Theorem 2.4 ([49]). VC−→p Y is a good categorial quotient:
(1) p is surjective.
(2) If W1 ⊂ VC ⊃ W2 are closed and GC-invariant, and W1 ∩W2 = ∅ then p(W1)∩p(W2) =∅.
(3) If W ⊂VC is closed and GC-invariant, then p(W)⊂Y is also closed.
(4) p is aGC-invariant morphism, i.e p(gv) =p(v)for all g ∈GC andv ∈VC. (5) For any open subsetU ⊂Y there is an isomorphism: C[U] p
∗
−→C[p−1(U)]GC, i.e p is a categorial quotient.
The pair (Y, VC −→p Y) is often simply called a GIT quotient and Y is denoted byY := VC
GC. One shall note by case (2) that ify∈Y, then there is a unique closed orbit inp−1(y) i.e GCv ⊂p−1(y). Thus one can think of Y as the set of all closed orbits of theGC-action, by the bijection: y7→GCv. Note that X
G is not necessarily the orbit space of the action, i.e p does not necessarily satisfy:
p(v1) =p(v2)⇔Gv1 =Gv2, ∀v1, v2 ∈VC.
Indeed this criterion would require that all the orbits of the action ofGare closed, however if this is true then such an action is said to be closed, and locally the ac- tion is always closed; in fact there always exist aG-invariant open subset U ⊂VC such that the restriction: U −−→p|U p(U), gives rise to a GIT quotient:
U, p|U
, that is also an orbit space ([49]).
We now recall some results of GIT over R used in this thesis from [7, 35, 39].
LetGC ⊂GL(EC) be a linearly complex reductive Lie group defined over R, and denote the real points: GC(R) :=GC∩GL(E), which is an algebraic real form. Now consider any closed Lie subgroup G ⊂ GC(R) containing the identity component GC(R)0 such that the Zariski-closure in GL(EC) is GC. Let GC act rationally on a complex vector space VC, and assume the action is defined over R. Thus the action restricts to a Lie group action of G on V. These are the assumptions of Richardson andSlowody [7]. A class of real Lie groups Gobtained in this way are for instance the class of real linear semisimple Lie groups which are fcc [39].
For such type of Lie groupsG, we have the following theorem:
Theorem 2.5 ([7]). Let G ⊂GL(E) be as above. Then the following statements hold:
(1) There exist a global Cartan involution of GL(E) leaving G invariant.
(2) If gl(E)−→θ gl(E) is a Cartan involution leaving g invariant, then Θ(G)⊂ G, where Θ is the global Cartan involution of GL(E) with differential θ.
(3) All Cartan involutions of Gare conjugate by an inner automorphism of G.
(4) Let G ρ
G
−→V GL(V) be a real representation. Then given any global Cartan involution Θof G, then there exist a global Cartan involutionΘ0 of GL(V) such that: ρGV(Θ(g)) = Θ0(ρGV(g)),∀g ∈G.
In this thesis we are mainly interested in the pseudo-orthogonal groups O(p, q), which are naturally real forms of O(p+q,C). Therefore throughout this section we can take as examples G := O(p, q) and GC := O(p+q,C). These groups are formally defined as follows:
Letg(−,−) be a real valued non-degenerate symmetric bilinear form (also called apseudo-inner product) on a real finite dimensional vector spaceV. Then the isom- etry group ofg(−,−), denotedO(V, g) is a real Lie group, and if (p, q) denotes the signature of g(−,−), then we define O(p, q) :=O(V, g). This group is semisimple forp+q≥3, and in general it is a real reductive algebraic group. Ifθ denotes the involution of V such thatg(·, θ(·)) is positive definite, then the involution:
O(p, q)3f 7→θf θ, is a Cartan involution of O(p, q). Thus
K :={f ∈O(p, q)|[θ, f] = 0} ⊂O(p, q),
is a maximally compact subgroup and one can show that K is isomorphic to the product of Lie groups: O(p)×O(q).
By complexifying g to gC, then (VC, gC) becomes a holomorphic inner product space such that the isometries: O(p+q,C), is a complex Lie group. By complex- ifying the real group O(p, q) by the map:
f 7→fC, f ∈O(p, q),
then O(p, q) becomes embedded as a real form ofO(p+q,C). In fact O(p+q,C) is the universal complexfication group of O(p, q), and it is a linearly complex reductive Lie group, which is semisimple forp+q≥3. Moreover sinceO(p+q,C)⊂ GL(VC) and O(p, q) ⊂ GL(V), then naturally the Zariski-closure of O(p, q) in GL(VC) is also precisely O(p+q,C).
If W ⊂ VC is a real form which is real-valued and positive definite on gC, say
˜ g :=g|C
W then the isometries of (W,g) denoted˜ O(p+q) is a compact real form of O(p+q,C).
Suppose G ρ
G
−→V GL(V) is any real representation, and g = k⊕p, is a Cartan decomposition with global decomposition: G = Kep, where K has Lie algebra k. Then one can choose a K-invariant inner product h·,·i onV such that dρGV(p) consists of symmetric operators w.r.t h·,·i.
One defines w.r.t ρGV and h·,·i:
Definition 2.6. A vector v ∈V is aminimal vector if||g·v|| ≥ ||v||for allg ∈G, where||v||2 :=hv, vi.
We denoteM(G, V)⊂V for the set of minimal vectors. As an example consider the adjoint representation G −→Ad GL(g), with G := O(p, q) (p+q ≥ 3). Then choosing a Cartan involution θ of g:=o(p, q) we may take ||v||2 :=−κ(v, θ(v)) as our inner product. Moreover it is straight forward to show that the minimal vectors are precisely those vectors v ∈ g satisfying [v, θ(v)] = 0. Thus if o(p, q) =k⊕p is the Cartan decomposition w.r.tθ, then k∪p⊂ M(G,g).
The following theorem relates the closure (w.r.t the classical topology) of a real orbit to the set M(G, V):
Theorem 2.7 (Richardson, Slodowy, [7]). The following statements hold:
(1) A real orbit Gv is closed if and only if Gv∩ M(G, V)6=∅.
(2) If v is a minimal vector then Gv∩ M(G, V) = Kv.
(3) If Gv is not closed then there exist p ∈ p such that etp·v → α ∈ V exist as t → ∞, and Gα⊂V is closed. MoreoverGα ⊂Gv is the unique closed orbit in the closure.
(4) A vector v ∈V is minimal if and only if
∀x ∈p
hx·v, vi = 0
, where x·v is the differential action dρGV(x)(v).
One shall note that the above theorem is also proved for a class of reductive Lie groups [29] that extends the class of groups Gin [7].
Let GC −→ρC GL(VC) denote (by assumption) the complexification of ρ i.e the following diagram commutes:
GC −−−→ρC GL(VC)
i
x
i x
G −−−→ρ GL(V) (2)
LetU be a compact real form ofGC such that the Lie algebra isk⊕ip, then one can choose (see [7]) a U-invariant Hermitian inner product H(·,·) on VC which is compatible withV, i.e
H(V, V)∈R, such that:
M(G, V)⊂ M(GC, VC).
Parts (1) and (2) of the above theorem is known as the Kempf-Ness Theorem for actions ρC. One shall also note that if G = U is a compact real form, then M(U, V) =V, sincep= 0 in this case.
The following theorem connects the real and complex case [7,35, 39]:
Theorem 2.8. The following statements are true:
(1) If v ∈V then a real orbit Gv ⊂V is closed (w.r.t the classical topology) if and only if GCv ⊂VC is closed.
(2) If v ∈V then GCv∩V =∪Nj=1Gvj for some natural number N ≥1.
In the case where G = U is a compact real form of GC then any real orbit Gv ⊂V is closed and the following special case of the previous theorem holds:
Theorem 2.9 ([7]). If G = U is a compact real form of GC then for any v ∈ V we have GCv∩V =Gv.
Part 3. Wick-rotations and real GIT
The following part is precisely the published paper in the Journal of Geometry and Physics:
Wick-rotations and real GIT, C. Helleland, S. Hervik, J. Geom. Phys. 123 (2018) 343-361, https://doi.org/10.1016/j.geomphys.2017.09.009.
Abstract. We define Wick-rotations by considering pseudo-Riemannian man- ifolds as real slices of a holomorphic Riemannian manifold. From a frame bundle viewpoint Wick-rotations between different pseudo-Riemannian spaces can then be studied through their structure groups which are real forms of the corre- sponding complexified Lie group (different real formsO(p, q) of the complex Lie groupO(n,C)). In this way, we can use real GIT (geometric invariant theory) to derive several new results regarding the existence, and non-existence, of such Wick-rotations. As an explicit example, we Wick rotate a knownG2-holonomy manifold to a pseudo-Riemannian manifold with split-G2 holonomy.
1. Introduction
In this paper we will study so-called Wick-rotations which were first used in physics as a mathematical trick relating Minkowski space to flat Euclidean space.
Here we will generalise the concept of Wick-rotations to more general pseudo- Riemannian geometries by considering the complexification to a holomorphic Rie- mannian manifold. The real pseudo-Riemannian manifolds will now manifest themselves as real slices of the complex holomorphic geometry. Utilizing this description we define Wick-rotations of pseudo-Riemannian manifolds as well as the stronger concept of astandard Wick rotation.
There are previous works considering complex geometry and Wick-rotations in different contexts [1,2,25,4,5,6]. Here, we will define the Wick-rotations based on observations made in [25] which is related to the definition of Wick-related spaces given in [6]. In fact, we adopt this definition but define the stronger concepts of Wick-rotations andstandard Wick-rotations. This enables us to connect the study of Wick-rotations to real GIT [35, 7] which recently has seen its appearence in the classification of pseudo-Riemannian geometries [8, 9]. Using old, as well as some new, results from real GIT we give several results regarding the possibily of Wick-rotating pseudo-Riemannian spaces to different signatures (see also [21]).
In this paper we will reserve the notion ofRiemannian space to the case when the metric is positive definite (of signature (+ +..+)). The Lorentzian case is the case of signature (−+ +..+). Note also the existence of the ”anti-isometry” which switches the sign of the metric: g 7→ −g. This anti-isometry induces the group isomorphism O(p, q)→ O(q, p) and hence our results are independent under this map.
2. Mathematical Preliminaries
Let E be a complex vector space. By considering only scalar multiplications by R ⊂ C, we can define a real vector space ER whose points are identical to E.
Multiplication with i inE defines an automorphism J :ER −→ER satisfying:
J◦J =−Id.
(3)
An endomorphism J satisfying eq.(3) is called a complex structure of ER. This correspondence between the complex vector space E and the real vector space ER equipped with the complex structureJ defines an isomorphism of these categories.
If we consider a complex vector subspace W ⊂ E, then its corresponding real vector space WR is a vector subspace of ER being invariant underJ. On the other hand, if W is a vector subspace of ER being invariant under J then we say that W is a complex subspace.
The linear map J can also be extended to the complexificationEC :=ER⊗RC. The complexification EC can then be split into the sum of the eigenspaces:
E1,0 ={u∈ER⊗RC
J u=iu}
and
E0,1 ={u∈ER⊗RC
J u =−iu}.
2.1. Real form of a complex vector space. A complex vector space E may be the complexification of a real vector space W:
E =WC:=W ⊗RC, in which case we will call W a real form of E.
Assume now thatW is a vector space overR. ThenW is naturally a real form of the complexification: WC, indeed the field extensionR,→Cinduces the inclusion W ,→WC, w 7→w⊗1. Furthermore, complex conjugation in C gives rise to an anti-linear involution ρ of WC:
ρ(w⊗z) =w⊗z.¯
The fixed-point set of ρ is W. Such a map is called a conjugation map of WC associated to W.
Some special examples of conjugation maps can be easily found among semi- simple complex Lie algebras. For examplesl(2,C) has real formssu(2) andsl(2,R) associated to the conjugation maps X7→ −X† and X 7→X¯ respectively.
Definition 2.1. Let E be a complex vector space with a complex structure J : ER −→ER. Then real linear subspaceW ⊂ERis called totally real ifW∩J(W) = 0.
In particular we see that if W is a maximal totally real subspace, then W is a real form ofE. We note that if W is totally real implies that the composition
WC:=W ⊗RC−→EC−→E1,0 ∼=E,
where the second map is the projection onto E1,0, is injective. This is important to us in the following as this implies that there is a connection between the com- plexification of real pseudo-Riemannian geometry and holomorphic Riemannian geometry. We note also that if W is a real form of E then the complex dimension of E and the real dimension ofW are equal.
2.2. Real slices.
Definition 2.2. Aholomorphic inner product is a complex vector spaceEequipped with a non-degenerate complex bilinear form g.
For a holomorphic inner product spaceE we can always choose an orthonormal basis. By doing so we can identifyE with Cn and the holomorphic inner product can be written as
g0(X, Y) =X1Y1+...+XnYn, (4)
whereX = (X1, ..., Xn) and Y = (Y1, ..., Yn).
Using this orthonormal basis it is also convenient to consider the group of trans- formations leaving the holomorphic inner product invariant. Consider a complex- linear map A :E −→ E. Using an orthonormal basis, we can represent the map by a complex matrix A:Cn −→Cn. Requiring thatg0(A(X), A(Y)) =g0(X, Y), for allX, Y, implies thatAtA=1. Consequently, the matrix Amust be a complex orthogonal matrix; i.e., A∈O(n,C).
Definition 2.3. Given a holomorphic inner product space (E, g). Then ifW ⊂E is a real linear subspace for which g
W is non-degenerate and real valued, i.e., g(X, Y)∈R, ∀X, Y ∈W, we will call W areal slice.
Some standard examples of real slices can be found by considering the holo- morphic inner product space (Cn, g0) with standard basis {e1, ..., en}. The real subspace:
W =Rnp := span{ie1, ..., iep, ep+1, ..., en}, (5)
is a real slice for any 0 ≤ p ≤ n. The restriction of g0 to W in this case is the standard pseudo-Euclidean metric with signature (p, n−p). Using the standard coordinateszk =xk+iyk forCn, we see that the restriction x1 =...=xp =yp+1 = ...=yn= 0 gives us the real slice Rnp.
Let us assume that W and fW are real slices of (Cn, g0). Consider the real slice W with real non-degenerate bilinear form h. By choosing a pseudo-orthonormal basis, we can write:
h(X, Y) =−X1Y1−...−XpYp+Xp+1Yp+1+...+XnYn, (6)
where X = (X1, ..., Xn) and Y = (Y1, ..., Yn) (real) for some p. This space has complexification WC by allowing X and Y to be complex n-tuples. By restricting g0(−,−) to WC, we see that WC is Cn in an orthonormal frame. Doing the same for Wf we note that (fW)C is also Cn in (possibly another) orthonormal frame.
However, since orthonormal frames are related by the action of the groupO(n,C), the real slices W and fW are related via the action of the group O(n,C) on Cn. Indeed, since any n-dimensional complex holomorphic inner product space (E, g) can be identified with (Cn, g0), any two real slices of E are related through the action of O(n,C) on E.
Definition 2.4. Let W ⊂ (E, g) be a real slice. We say an involution W −→θ W, is a Cartan involution of W, if gθ(·,·) :=g
W(·, θ(·)), is an inner product onW. Of course Cartan involutions always exist as linear maps, and the definition generalises the notion of a Cartan involution of a semi-simple Lie algebra. Indeed special examples can be found within semi-simple real forms g of a complex semi- simple Lie algebra gC, which are real slices w.r.t the holomorphic Killing form:
−κ(−,−), on the complex Lie algebra gC. So a Cartan involution: g −→θ g, will in this case in fact be an involution of Lie algebras, and will be unique up to conjugation by inner automorphisms of g. Explicit examples of real slices are the pseudo-orthogonal real Lie algebras o(p, q) of o(n,C) with signatures
p 2
+
q 2
,2pq .
It is also important to note the following (see e.g., [6]):
Proposition 2.5. The real slices of a holomorphic inner product space are totally real subspaces.
2.3. Compatible real forms. Associated to any real formW of a complex vector space E ∼= WC we know that there is a conjugation map E −→σ E with fix points W. The space E may have another real form Wf, also with a conjugation map ˜σ which fixes pointwise Wf. So we have the notion of compatibility among two real forms in the following definition:
Definition 2.6. The two real forms W and fW of E are said to be compatible if their conjugation maps commute, i.e [σ,σ] = 0.˜
For two compatible real forms W and Wf of E we may write:
W = (W ∩fW)⊕(W ∩ifW) and Wf= (W ∩fW)⊕(fW ∩iW).
In the case of Lie algebras the real forms will have conjugation maps which are also real Lie homomorphisms. As an example consider the real forms o(p, q) and o(˜p,q) embedded into˜ o(n,C) with n = p+q = ˜p+ ˜q, with corresponding
conjugation maps:
X 7→ −Ip,qXI¯ p,q, X 7→ −Ip,˜˜qXI¯ p,˜˜q,
whereIp,q := (aij) is the n×n diagonal matrix with entries: aii= 1 for 1≤i≤p, and aii = −1 for p + 1 ≤ i ≤ n. It is easy to see that o(p, q) is compatible with o(˜p,q), and also observe that a Cartan involution for both real forms may˜ be chosen to be X 7→ X, i.e the Cartan involutions also commute, and we may¯ choose the compact real form:
o(n) ={X ∈o(n,C)|X = ¯X},
which will be compatible with botho(p, q) and o(˜p,q). This means that if˜ o(p, q) =t⊕p, o(˜p,q) = ˜˜ t⊕˜p,
denotes the Cartan decompositions then we have:
o(n) =t⊕ip= ˜t⊕i˜p.
We shall refer to such a triple:
o(p, q),o(˜p,q),˜ o(n)
, as a compatible triple of real forms. We define this not only for Lie algebras, but also for general real slices of the same dimension:
Definition 2.7. LetW and fW be real slices of (E, g). Assume they are both real forms of WC ⊂ (E, g). Let V be another real slice of E, and a real form of WC, with Euclidean signature. Suppose W,Wf and V are pairwise compatible, then a triple: (W,W , Vf ), will be called a compatible triple.
Note that a compatible triple (W,fW , V), implies that we may choose Cartan involutions ofW,WfandV which commute, this is by construction. We shall often refer to V as a compact real slice of E. In the case of W ∩W˜ = 0, we note that W = iW˜, so this corresponds to an anti-isometry, i.e., changing the metric from: g 7→ −g, a standard example is the compatible triple:
iR⊕R,R⊕iR,R2
, in (C2, g), with g(−,−) the standard holomorphic inner product. However there exist compatible triples not of this form, i.e., with W ∩W˜ 6= 0, and to find such examples, it is sufficient to look at compatible triple of Lie algebras. In fact, we may say something stronger in the case of a compatible triple of semi-simple Lie algebras.
Indeed we now show that if we have a compatible triple of semi-simple Lie algebras (g,˜g,u) withucompact like in the example above, then the compact/non- compact parts of the Cartan decompositions of the real forms must intersect. We denotet (respectively ˜t) for the compact part, andp (respectively ˜p) for the non- compact part. This is clear ifg= ˜g, so assume they are not equal nor isomorphic.
Proposition 2.8. Assume g˜g. We have t∩˜t6= 0, and if none of the real forms are compact and they are both simple then also p∩p˜6= 0.
Proof. We may assume none of the real forms are compact, because then the first part is trivial. Suppose that t∩˜t = 0, then it is easy to check that t ⊂ i˜p and
˜t⊂ip. Indeed if x∈tthen becauseg∩˜g=p∩˜pthenx=p+i(˜t+ ˜p) for suitable p,p˜∈˜pand ˜t∈˜t. But then,
x−ip˜=p+i˜t ∈u∩iu,
and consequently x = ip. The case ˜˜ t ⊂ ip is similar. But then [t,t] ⊂ ˜t∩t, i.e must be zero, and similarly [˜t,˜t] ⊂ ˜t∩t. So we conclude that ˜t and t must be abelian. Now the only simple Lie algebra with abelian compact part issl(2,R), i.e it follows that
g∼= ˜g∼=⊕kjsl(2,R),
for a suitable k. But since we assume g and ˜g are non-isomorphic, then this is a contradiction. Now for the second statement suppose p ∩˜p = 0. Then one easily checks that ˜p ⊂ it, and it is a standard result that [˜p,˜p] = ˜t and [p,p] =t using that the real forms are simple, and so therefore ˜t ⊂ [it, it] ⊂ t. Of course we similarly must have p ⊂ i˜t, so we conclude that t = ˜t. Hence g∩g˜ = t = ˜t.
However this will require ˜p ⊂ it = i˜t, proving that ˜p = 0. Hence ˜g is compact, which contradicts our assumptions. The proposition is proved.
3. Holomorphic Riemannian manifolds
3.1. Complexification of real manifolds. We will now consider the case where we have a real pseudo-Riemannian manifold. The aim is to consider analytic continuations of such via its complexification and it is thus necessary to assume that the manifold is analytic. We will also assume that the real dimension of the real manifold, and the complex dimension of the complex manifold are equal (unless stated otherwise).
Let us first start with a few definitions (see [6]).
Definition 3.1. Given a complex manifold M with complex Riemannian metric g. If a submanifold N ⊂ M for any pointp∈ N we have that TpN is a real slice of (TpM, g) (in the sense of Defn. 2.3), we will call N a real slice of (M, g).
This definition implies that the induced metric from M is real valued on N. N is therefore a pseudo-Riemannian manifold. This further implies that real slices are totally real manifolds.
We will also define the notion of Wick-related spaces, Wick-rotated spaces, as well as a standard Wick-rotation.
Definition 3.2 (Wick-related spaces). Two pseudo-Riemannian manifoldsP and Q are said to be Wick-related if there exists a holomorphic Riemannian manifold (M, g) such that P and Q are embedded as real slices of M.
Wick-related spaces was defined in [6]. However, we also find it useful to define:
Definition 3.3 (Wick-rotation). If two Wick-related spaces intersect at a point p in M, then we will use the term Wick-rotation: the manifold P can be Wick- rotated to the manifold Q(with respect to the point p).
Definition 3.4 (Standard Wick-rotation). Let the P and Q be Wick-related spaces having a common point p. Then if the tangent spaces TpP and TpQ are embedded: TpP, TpQ ,→(TpP)C ∼= (TpQ)C,→TpM such that they form a compat- ible triple, then we say that the spaces P and Q are related through a standard Wick-rotation.
We note in the case where P and Q are Wick-rotated by a standard Wick- rotation, andQis a real slice of Euclidean signature (i.e., it is a Riemannian space), then the tangent spaces: TpP and TpQ, can be embedded into (TpP)C ∼= (TpQ)C, such that they are compatible real forms. Also in the case where both real slices:
P andQ, are Wick-rotated of the same signatures, then they are also Wick-rotated by a standard Wick-rotation. Indeed we can identify TpP ∼= TpQ (as symmetric non-degenerate bilinear spaces), and in this case the real slices will be compatible with each other (since they are equal as sets in (TpP)C). Moreover there is a natural compact real slice: W ⊂(TpP)C, which is compatible withTpP.
The following proposition is immediate by definition of a compatible triple:
Proposition 3.5. Two Wick-rotated spaces P andQ by a standard Wick-rotation gives rise to Cartan involutions of TpP and TpQ which commute.
These three definitions are of increasing speciality; Wick-related spaces need not intersect at a pointp; nor is there a guarantee that Wick-rotated spaces have commuting Cartan involutions. This all depends on the way the real forms are imbedded into the complexification O(n,C).
However, in physics, all examples of Wick-rotations (known to the authors) are standard Wick-rotations in the sense above.
3.2. Complex differential geometry. It is useful to review some of the results from complex differential geometry especially in the holomorphic setting.
A complex Riemannian manifold is a complex manifold M equipped with a symmetric, C-bilinear, non-degenerate form g. A vector field is holomorphic if and only if it has holomorphic component functions with respect to any local com- plex coordinates. The holomorphic tangent bundleT M, can be constructed using the construction E1,0 via the complexification of T M. Similarly, a tensor field T over the holomorphic tangent bundle is holomorphic if and only if the component functions Tµ1...µlν1...νk are holomorphic with respect to any local holomorphic co- ordinates {z1, ..., zn} on M. Note also that the sum or tensor multiplication of two holomorphic tensors are holomorphic, so is the contraction of a holomorphic tensor.
For any complex Riemannian manifold there is a unique Levi-Civita connection
∇ (just as in pseudo-Riemannian case) satisfying:
1. [X, Y] =∇XY − ∇YX (torsion-free), (7)
2. ∇Xg = 0, (metric compatible) (8)
for all vector fields X and Y.
For a holomorphic metric, the Levi-Civita connection ∇is also holomorphic (as follows from the Koszul equations), so is the Lie bracket. This implies that the holomorphic Riemann curvature tensor,
R(X, Y)Z =∇X∇YZ− ∇Y∇XZ− ∇[X,Y]Z, (9)
is also holomorphic. Hence, for a holomorphic metric, the connection, and all the curvature tensors inherit this property: they are all holomorphic.
We remark that this implies that the standard equations for computing the con- nection coefficients, Riemann curvature tensors etc. which are known from the pseudo-Riemannian case can be used more or less unaltered for a holomorphic Riemannian manifold. Furthermore, this has profound consequences for us as the complexification of a real pseudo-Riemannian manifold is a holomorphic Riemann- ian manifold. Thus, given a real slice N ⊂M then the curvature tensors of N are uniquely extended to a neighbourhood of N inM.
We also remark that since contractions, tensor products preserve holomorphy, polynomial curvature scalars (as considered in [25]) are also holomorphic and is uniquely determined by knowing them on a real slice. Consequently, we get the following result: If a pseudo-Riemannian space N is obtained from a pseudo- Riemannian space P by Wick-rotation w.r.t. a point p, then their polynomial curvature invariants match at p.
Thus, we note the following important facts (M is the ambient holomorphic complex Riemannian manifold) :
(1) For two Wick-related spaces, all the Riemannian curvature tensors can be obtained fromM by restricting to the real slices.
(2) If a curvature tensor is identically zero for a pseudo-Riemannian manifold, N, then it is identically zero in a neighbourhood of N in M.
3.3. Real slices from a frame-bundle perspective. In the frame-bundle for- mulation of differential geometry, the Riemannian case is a frame-bundle with an O(n) structure group. In general, the pseudo-Riemannian case, has a O(p, q) structure group. As we saw earlier, holomorphic Riemannian geometry hasO(n,C) structure group. The relation between the real slices (with structure groupO(p, q)) and the holomorphic Riemannian case is related through the complexification of O(p, q)C∼=O(n,C).
In the Wick-rotated case, at the intersection point the different real slices with structure groups O(p, q) and O(˜p,q) will both be embedded in˜ O(n,C). Indeed if P and Q are Wick-rotated at a point p ∈ M of the same real dimension, say
n, then TpP and TpQ are real slices of TpM, of say signatures (p, q) and (˜p,q)˜ respectively. Now since these are totally real spaces, we have natural embeddings of
(TpP)C,→TpM, and (TpQ)C,→TpM.
We can restrict the metric g onTpM to (TpP)C and (TpQ)C so they become holo- morphic inner product subspaces of TpM. In particular since we have an isomor- phism: (TpQ)C −→ψ (TpP)C (as holomorphic inner product spaces), then we have natural embeddings:
TpP, TpQ ,→(TpP)C⊂TpM,
as real slices of (TpP)C, with their signatures: (p, q) and (˜p,q), respectively. More-˜ over when restricting to (TpP)C on the holomorphic metric g onM, gives another holomorphic inner product, with structure group:
O(n,C) :={(TpP)C −→f (TpP)C|g(f(−), f(−)) =g(−,−)}.
The pseudo-orthogonal groups: O(p, q) (structure group of P) and O(˜p,q) (struc-˜ ture group of Q), will now be embedded as real forms via ψ into O(n,C).
A tensor x over the point p ∈ P ∩Q w.r.t P will therefore be considered as a vector x∈V for some appropriate vector space, and similarly a tensor ˜x w.r.t to Q over the same point p will be in another real form ˜V ⊂ VC. This could be, for example, the Riemann tensor or covariant derivatives of the Riemann tensor, restricted to the point. If the two spaces are Wick-rotated the orbitsGx, and ˜G˜x where G:=O(p, q) and ˜G:=O(˜p,q), are embedded into the same complex orbit˜ GCx∼=GCx, for˜ GC :=O(n,C). Hence, we have the two embeddings:
O(n,C)·x O(n,C)·x˜ x
x
O(p, q)·x O(˜p,q)˜ ·x˜
A necessary condition for the existence of Wick-rotated x and ˜x is therefore the existence of a complex orbit in which both real orbits are embedded. In the case of a standard Wick-rotation we know that the tangent spaces TpP and TpQ form a compatible triple with a compact real slice, say, W (a real form of (TpP)C of Euclidean signature w.r.t g), when embedded into (TpP)C ⊂ TpM. So w.r.t W, there is a compact real form: O(n), also embedded inO(n,C) (as above). Denote nowo(p, q),o(˜p,q) and˜ o(n), for the real forms (of Lie algebras) ofO(p, q), O(˜p,q)˜ and O(n) respectively, embedded into o(n,C) (the Lie algebra of O(n,C)) w.r.t a standard Wick-rotation. Then we have the following observation:
Lemma 3.6.The triple of real forms:
o(p, q),o(˜p,q),˜ o(n)
, embedded intoo(n,C) under a standard Wick-rotation is also a compatible triple of Lie algebras.
